And where would we find the resources to build something like that? Let alone move it to orbit.
On tiny scales, carbon bonds have sufficient strength/weight ratio so that the elevator might support it's own weight, but there's not that much margin left. As the macrostructure likely requires a multiple of weight, to account for the fact that a kilometers-long string can't be manufactured perfect to the single atom, and that damage can and will occur (e.g. any radiation), etc, etc - according to our current knowledge of physics it might as well be that a sufficiently strong macroscale material for an Earth-sized space elevator cannot exist. Moon or Mars is a different issue, though.